Construction of Fischer's sporadic group Fi'_{24} inside GL_{8671}(13)

TL;DR

This paper constructs a specific sporadic simple group Fi'_{24} within a large general linear group over GF(13), providing explicit representations, character tables, and subgroup structures, advancing the computational understanding of this complex group.

Contribution

It offers a novel explicit construction of Fi'_{24} inside GL_{8671}(13) using known subgroups and algorithms, including permutation representations and subgroup analyses.

Findings

Constructed Fi'_{24} inside GL_{8671}(13)

Determined a faithful permutation representation of degree 306936

Identified conjugacy classes of involutions and their centralizers

Abstract

In this article we construct an irreducible simple subgroup G = <q, y, t, w> of GL_{8671}(13) from an irreducible subgroup T of GL_{11}(2) isomorphic to Mathieu's simple group M_{24} by means of Algorithm 2.5 of [13]. We also use the first author's similar construction of Fischer's sporadic simple group G_1 = Fi_{23} described in [11]. He starts from an irreducible subgroup T_1 of GL_{11}(2) contained in T which is isomorphic to M_{23}. In [7] J. Hall and L. S. Soicher published a nice presentation of Fischer's original 3-transposition group Fi_{24} [6]. It is used here to show that G is isomorphic to the simple commutator subgroup Fi'_{24} of Fi_{24}. We also determine a faithful permutation representation of G of degree 306936 with stabilizer G_1 = <q, y, w> $\cong$ Fi_{23}. It enabled MAGMA to calculate the character table of G automatically. Furthermore, we prove that G has two conjugacy classes of involutions z and u such that C_G(u) = <q, y, t> $\cong$ 2Aut(\Fi_{22}). Moreover, we determine a presentation of H = C_G(z) and a faithful permutation representation of degree 258048 for which we document a stabilizer.